Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #10593098⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×((C22⋊C9)⋊C2)⋊C3 |
| commutator, cosocle, hypocenter | Q8⋊C9 | C23×C3 |
| Frattini, socle | C2×C3 | C22×C3×S4 |
| Fitting | C2×C32×#41 | D3 |
| hypercenter | C2×C4 | ((C22⋊C9)⋊C2)⋊C3 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
540 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^3,3^3,2^3|b^2=i,e^3=f,g^2=i,h^2=i,c^a=ci,e^a=e^2f^2,f^a=f^2,g^a=gh,h^a=hi,e^d=ef,g^e=h,h^e=ghi,h^g=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72245649, #72245749, #72245974 |
| maximal normal subgroup of | #72245649, #72245749, #72245974 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30489);